The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  X
 0  X 2X+2 X+2  0 X+2 2X+2 3X  0 X+2 3X 2X+2  0 X+2 2X+2  X  0 X+2 2X+2 3X  0 X+2 2X+2  X  0 X+2 2X+2  X  0 X+2 2X+2 3X 2X 3X+2  2 3X 2X 3X+2  2  X 2X 3X+2  2  X 2X 3X+2  2 3X 2X 3X+2  2 3X 2X 3X+2  2 3X 3X+2 2X  2  X 2X 3X+2  2  X  0 X+2 2X+2 3X X+2 X+2
 0  0 2X  0  0  0 2X  0  0 2X 2X 2X  0 2X 2X 2X 2X  0  0 2X 2X 2X  0  0 2X  0  0 2X 2X 2X  0  0 2X 2X 2X 2X 2X 2X 2X 2X  0  0  0  0  0  0  0  0 2X 2X 2X 2X 2X 2X 2X  0  0  0  0 2X  0  0  0  0  0  0  0  0 2X 2X
 0  0  0 2X  0  0  0 2X 2X 2X 2X 2X 2X  0 2X  0 2X  0  0 2X 2X 2X  0  0  0 2X 2X  0  0  0 2X 2X  0  0  0  0 2X 2X 2X 2X 2X 2X 2X 2X  0  0  0  0  0  0  0  0 2X 2X 2X  0 2X 2X 2X 2X  0  0  0 2X  0  0  0 2X 2X  0
 0  0  0  0 2X 2X 2X 2X 2X  0 2X  0  0 2X 2X  0  0  0  0  0 2X 2X 2X 2X 2X 2X 2X 2X  0  0  0  0  0  0 2X 2X 2X 2X  0  0  0  0 2X 2X 2X 2X  0  0 2X 2X  0  0  0  0 2X 2X 2X 2X  0 2X  0  0 2X  0  0  0 2X 2X  0  0

generates a code of length 70 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+87x^68+128x^69+592x^70+128x^71+86x^72+1x^76+1x^136

The gray image is a code over GF(2) with n=560, k=10 and d=272.
This code was found by Heurico 1.16 in 0.328 seconds.